Solutions of the 3D Navier-Stokes equations for initial data in Ḣ1/2: Robustness of regularity and numerical verification of regularity for bounded sets of initial data in Ḣ1

We consider the three-dimensional Navier-Stokes equations on a periodic domain. We give a simple proof of the local existence of solutions in Ḣ1/2, and show that the existence of a regular solution on a bounded time interval [0,T] is stable with respect to perturbations of the initial data in Ḣ1/2 and perturbations of the forcing function in L2(0,T;H−1/2). This forms the key ingredient in a proof that the assumption of regularity for all initial conditions in any given ball in Ḣ1 can be verified computationally in a finite time, strengthening a previous result of Robinson and Sadowski [J.C. Robinson and W. Sadowski, Numerical verification of regularity in the three-dimensional Navier-Stokes equations for bounded sets of initial data, Asymptot. Anal. 59 (2008) 39-50].